Create the page "No u" on this wiki! See also the search results found.

• Cauchy-Schwarz Inequality
  ...bf{u}+\gamma \mathbf{v})</math> and <math>\mathbf{q} = \frac{1}{2}(\mathbf{u}-\gamma \mathbf{v})</math>. Importantly, we have ...le \mathbf{u},\gamma \mathbf{v}\rangle + \langle \gamma \mathbf{v},\mathbf{u}\rangle - \gamma \bar{\gamma}||\mathbf{v}||^2)
  13 KB (2,048 words) - 15:28, 22 February 2024
• Constructive counting
  ...that each entry is between <math>2</math> and <math>9</math> inclusive and no two consecutive entries are equal?'' ...luding <math>2</math> and <math>9</math>; the key restriction here is that no two boxes right next to each other can have the same number.
  12 KB (1,896 words) - 23:55, 27 December 2023
• Fundamental Theorem of Calculus
  ...me <math>{t}</math> is <math>e^{t^2}</math> meters/second. (Yes, probably no object really moves this way, but just pretend.) ''Approximately'' how far <cmath>L(f, P) \le g(b) - g(a) \le U (f, P)</cmath>
  11 KB (2,082 words) - 15:23, 2 January 2022
• Mock AMC
  ...ular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but ! scope="row" | '''Mock AMC U'''
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2004 AMC 12A Problems
  ...at-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters? Let <math>U=2\cdot 2004^{2005}</math>, <math>V=2004^{2005}</math>, <math>W=2003\cdot 20
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2005 AIME I Problems/Problem 13
  # There are no right angle turns in the particle's path. ...mber of diagonals. Let <math>R</math> represent a move to the right, <math>U</math> represent a move upwards, and <math>D</math> to be a move that is di
  5 KB (897 words) - 00:21, 29 July 2022
• 2000 AIME I Problems
  Find the least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two posit ...s. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 1983 AIME Problems/Problem 15
  ...ath>3</math>. By considering the graph of <math>y=f(x)</math>, which is a "U-shaped" parabola, it is now evident that <math>f(-3) > 0</math> and <math>f === Solution 7 (No Trig) ===
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1988 AIME Problems/Problem 15
  ...ion, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.) <math>\textbf{Case 1:}</math> Since letter 9 arrived before lunch, no further letters will arrive, and the number of possible orders is simply th
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 13
  ...ath>c_{i}</math> is some coefficient. However, since <math>F(x)</math> has no <math>x</math> term, it must be true that <math>c_{15} = 1</math>. ...>g(x)</math> are also roots of <math>f(x)</math>. Let these roots be <math>u,v</math>. We get the system
  10 KB (1,585 words) - 03:58, 1 May 2023
• 2014 USAJMO Problems
  ...which <math>A</math> cannot win in a finite number of moves, or prove that no such minimum value exists. ...erline{CA}</math> and <math>\overline{AB}</math>, respectively. Let <math>U,V</math> be the intersections of line <math>EF</math> with line <math>MN</m
  3 KB (600 words) - 16:42, 5 August 2023
• 2002 AIME I Problems/Problem 9
  * Ulysses starts with the third picket and paints every <math>u</math>th picket. ...itive integer <math>100h+10t+u</math> paintable when the triple <math>(h,t,u)</math> of positive integers results in every picket being painted exactly
  4 KB (749 words) - 19:44, 25 April 2024
• Holomorphic function
  Specifically, let <math>u, v : \mathbb{R \times R \to R}</math> be definted <cmath> u(x,y) = \text{Re}\,f(x+iy), \qquad v(x,y) = \text{Im}\,f(x+iy) . </cmath>
  9 KB (1,537 words) - 21:04, 26 July 2017
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...se variables, which does uniquely determine these variables - but there is no obvious way of computing them. We will show a different solution. ...th>d_2</math> one can always uniquely determine the coefficients <math>s,t,u</math>.
  3 KB (568 words) - 15:50, 3 April 2012
• 1977 Canadian MO Problems
  ...ath>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> Let <math>0<u<1</math> and define
  3 KB (560 words) - 19:23, 10 March 2015
• 2007 iTest Problems
  \text{(L) There is no such value of } r\qquad</math> \text{(U) }\frac{-2007}{2^{14}}\qquad</math>
  33 KB (5,177 words) - 21:05, 4 February 2023
• 2007 AMC 12A Problems/Problem 22
  Case 1: <math>196u \Longrightarrow u = 9</math>. Easy to directly disprove. ...S(n)) = 8+u</math> if <math>u \le 2</math> and <math>S(S(n)) = 2 + (u-3) = u-1</math> otherwise.
  15 KB (2,558 words) - 19:33, 4 February 2024
• 1960 IMO Problems/Problem 6
  ...h>u = \sqrt{x^2+1}</math>. Since <math>x > 0</math>, it follows that <math>u > 1</math>. We now have: <cmath>W_1 = (u + 1)^3</cmath>
  7 KB (1,214 words) - 18:49, 29 January 2018
• Graph (graph theory)
  ...simple, loopless'' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For grap ...<math>v</math> is ''isolated'' if <math>d(v) = 0</math>, i.e. if there are no edges incident to <math>v</math>.
  8 KB (1,428 words) - 10:26, 27 August 2020
• 1997 PMWC Problems
  ...</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and label("$U$",(-1/6,sqrt(3)/3),NW);
  15 KB (2,057 words) - 19:13, 10 March 2015
• 1994 USAMO Problems
  Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> ...e of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modi
  2 KB (391 words) - 07:58, 19 July 2016
• 2007 AMC 12B Problems/Problem 24
  Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>. ...ring the denominator and simplifying, we get a quadratic in terms of <math>u</math>:
  9 KB (1,522 words) - 22:46, 12 May 2022
• 1984 IMO Problems/Problem 4
  ...ath>, and with all conditions satisfied. We shall prove that there exists no point <math>T</math> on <math>CD</math> such that <math>T</math> is a midpo ...the circle with diameter <math>CD</math> touching <math>AB</math> at <math>U</math>(as well as the circle with diameter <math>AB</math> touching <math>C
  4 KB (750 words) - 23:49, 29 January 2021
• 1991 IMO Problems/Problem 4
  ...ge joining <math>v_i</math> and <math>v_{i+1}</math> with the number <math>u+i-1</math>. ...that vertex will be sorrounded by edges whose greates common factor is 1, no matter how the other edges sorrounding it are labelled. Notice that from o
  4 KB (668 words) - 17:45, 30 January 2021
• 2002 AMC 10B Problems
  ...<math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in a ...ath> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?
  10 KB (1,540 words) - 22:53, 19 December 2023
• 1974 USAMO Problems
  ...her and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: ...ngle ADB = \angle BDC = \angle CDA = 120^\circ</math>. Prove that <math>x=u+v+w</math>.
  3 KB (427 words) - 18:55, 3 July 2013
• 2007 AMC 10B Problems/Problem 25
  ...th>(a,b)</math> are there such that <math>a</math> and <math>b</math> have no common factors greater than <math>1</math> and: Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>.
  8 KB (1,425 words) - 23:05, 21 June 2023
• 2010 AIME I Problems/Problem 13
  ...ath>, and segment <math>CD</math> at distinct points <math>N</math>, <math>U</math>, and <math>T</math>, respectively. Line <math>\ell</math> divides re label("$U$",(87.13819,5.00321),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• 2011 AIME I Problems/Problem 8
  ...math> on <math> \overline{CY} </math>, and points <math>Z</math> and <math>U</math> are on <math> \overline{AB} </math> with <math>Z</math> on <math> \o pair[] A, B, C, U, V, W, X, Y, Z;
  6 KB (1,077 words) - 21:47, 12 April 2022
• 1999 AMC 8 Problems
  label("U",(10,-2),S); The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is <math>11/4</math>. To the nearest who
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2011 USAMO Problems/Problem 3
  In hexagon <math>ABCDEF</math>, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy <math>\ang ...}</cmath> <cmath>\vec{w} = \vec{CD} + \vec{FA}</cmath> Clearly, <math>\vec{u}+\vec{v}+\vec{w}=\textbf{0}</math>.
  11 KB (1,925 words) - 12:07, 31 August 2023
• Mock Amc 12
  ...ular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but ! scope="row" | '''Mock AMC U'''
  18 KB (2,206 words) - 19:41, 24 December 2020
• 2000 AMC 8 Problems/Problem 11
  Casework by the units digit <math>u</math> will help organize the answer. <math>u=0</math> gives no solutions, since no real numbers are divisible by <math>0</math>
  2 KB (382 words) - 15:59, 1 January 2024
• 2011 IMO Problems/Problem 2
  ...S}</math> be a finite set of at least two points in the plane. Assume that no three points of <math>\mathcal S</math> are collinear. A ''windmill'' is a (By u/basuboss on reddit)
  3 KB (577 words) - 23:19, 26 November 2023
• 1950 AHSME Problems
  ...igit number whose tens' digit is <math>t</math>, and units' digit is <math>u</math>, the new number is: ...extbf{(B)}\ 100t+10u+1\qquad\textbf{(C)}\ 1000t+10u+1\qquad\textbf{(D)}\ t+u+1\qquad\\ \textbf{(E)}\ \text{None of these answers} </math>
  22 KB (3,306 words) - 19:50, 3 May 2023
• BBCode:Tutorial
  ...utton is for underlining text. When you click on it, the symbols [u] and [/u] appear. Then you can type in whatever you want to be underlined: ...tags (not implemented on the top BBCode bar in the forums). There will be no grey box that there is when using [code][/code], and it will not automatica
  10 KB (1,680 words) - 19:26, 5 April 2023
• 2002 Romanian NMO Problems
  has no positive integer solutions. ...tarrow\mathbb{R}</math> be a function that has limits at any point and has no local extrema. Show that:
  10 KB (1,695 words) - 10:03, 10 May 2012
• 1955 AHSME Problems
  <math> \textbf{(A)}\ \text{no real values of }x\qquad\textbf{(B)}\ \text{either }x=1\text{ or }x=2\qquad\ The negation of the statement "No slow learners attend this school" is:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 2020 Mock Combo AMC 10
  How many permutations of the string <math>AABBCC</math> are there such that no two identical letters are adjacent to each other? ...positive integers <math>s, t, u, v, w, x, y, z</math> where <math>s > t > u > v > w > x > y > z</math>.
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2013 AMC 12A Problems/Problem 25
  ...part of a radical of a complex number: <math>\sqrt{u}</math>, where <math>u = v+wi = r e^{i\theta}</math>. <math>\operatorname{Im}(\sqrt{u}) = \operatorname{Im}(\pm \sqrt{r} e^{i\theta/2}) = \pm \sqrt{r} \sin(\thet
  4 KB (766 words) - 14:22, 8 October 2023
• 2013 AMC 12B Problems
  ...rn a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the e for (real u = 0; u < len; u += step){
  16 KB (2,459 words) - 02:46, 30 January 2021
• 2013 USAJMO Problems/Problem 6
  ...x - 9</math>. It is easily verified that <math>x^3 - 8x + 9 = 0</math> has no solution in positive numbers greater than 1. Thus, <math>\sqrt{x + xyz} \ge ...e can let <math>u = \sqrt{(y-1)(z-1)}</math>! Thus, <cmath>u^2 - 2u + 2 = (u-1)^2 + 1 > 0</cmath> and the claim holds for x = 1.
  3 KB (517 words) - 20:02, 30 April 2014
• Northeastern WOOTers Mock AIME I Problems
  ...e{uv.xy}+\overline{xy.uv} </cmath> is an integer? As an example, if <math>(u,v,x,y)=(0,1,2,3)</math>, then we have <math> N = 1.23 + 23.01 = 24.24 </mat ...re is no overlap. He continues to place T-Polyominos randomly until he can no longer do so. Let the probability that he will cover the entire board be <m
  8 KB (1,336 words) - 09:10, 30 May 2020
• 1983 IMO Problems/Problem 4
  ...olor on the respective side must have the opposite color if we are to have no monochromatic right-angled triangles. ...the points <math>U,V,T</math> must have different colors if we are to have no monochromatic right-angled triangle, and this is impossible (we only have t
  2 KB (373 words) - 23:40, 29 January 2021
• 2013 IMO Problems/Problem 2
  ...sts of <math>2013</math> red points and <math>2014</math> blue points, and no three of the points of the configuration are collinear. By drawing some lin *no line passes through any point of the configuration;
  3 KB (643 words) - 01:31, 19 November 2023
• 1973 AHSME Problems
  has no solution if and only if <math>n</math> is equal to ...alf at <math>v</math> miles per hour. Car B travels half the time at <math>u</math> miles per hour and half at <math>v</math> miles per hour. The averag
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1988 AHSME Problems
  ...=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5)); draw(X--Y--Z--W--X^^T--V--X^^Y--U);
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1972 AHSME Problems
  \textbf{(D) }\text{no }y\qquad real t=pi/8;real u=7*pi/12;real v=13*pi/12;
  17 KB (2,732 words) - 13:54, 20 February 2020
• 2007 iTest Problems/Problem 24
  \text{(U) }20 \quad ...2</math> in <math>N</math> is at least <math>1</math>, but since there are no powers of <math>2</math> in <math>2007</math>, the exponent of <math>2</mat
  2 KB (340 words) - 19:49, 30 June 2018
• 1964 AHSME Problems
  ...uotient is <math>u</math> and the remainder is <math>v</math>, where <math>u</math> and <math>v</math> are integers. \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1963 AHSME Problems
  \textbf{(E)}\ \text{no real values of }y </math> Between <math>B</math> and <math>M</math> point <math>U</math> is taken,
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1961 AHSME Problems
  ...the squares of the positive integers be called <math>u</math>; thus <math>u</math> is the set <math>1, 4, 9, 16 \ldots</math>. we say that the set is closed under that operation. Then <math>u</math> is closed under:
  18 KB (2,905 words) - 18:33, 5 April 2023
• 2013 UNCO Math Contest II Problems
  Three hours; no electronic devices. Justify your answers. Clear and concise presentations w <math>H = 34</math>. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail st
  11 KB (1,648 words) - 09:55, 20 December 2021
• 2015 AMC 12A Problems/Problem 23
  Opposite side: Probability is obviously <math>1</math>, no matter what. If <math>\sin(u)=2n</math>, then <math>dn=\frac{\cos(u)}{2}\;du</math> (differentiate both sides).
  12 KB (1,981 words) - 18:33, 3 September 2023
• 2015 AIME I Problems/Problem 10
  .... Since only the absolute value of <math>f(0)</math> is required, there is no loss of generalization by stating that <math>f(1)=12</math>, and <math>f(2) ...th>a\neq 0</math> (else <math>f</math> is not cubic) where <math>\{q,r,s,t,u,v\}</math> is the same as the set <math>\{1,2,3,5,6,7\}</math>. Subtracting
  8 KB (1,474 words) - 10:00, 10 November 2023
• 2015 AMC 8 Problems
  ...four symbols. The first must be a vowel (<math>A, E, I, O,</math> or <math>U</math>), the second and third must be two different letters among the <math ...ber of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If <math>\tfrac{1
  16 KB (2,322 words) - 14:04, 2 February 2024
• 2016 AMC 12B Problems/Problem 20
  ...team won <math>10</math> games and lost <math>10</math> games; there were no ties. How many sets of three teams <math>\{A, B, C\}</math> were there in w
  3 KB (571 words) - 16:29, 15 July 2023
• 2016 AIME I Problems/Problem 15
  ...AYB</math> is a parallelogram. Let <math>u:=ZX</math>, so that <math>ZY=ZF=u+47</math>. ...-47\right).</cmath> Now compute <cmath>AB^{2}=4AZ^{2}=4\cdot ZX\cdot ZY=4u(u+47)=37\cdot 67-47^{2}=\textbf{270}.</cmath>
  14 KB (2,427 words) - 17:12, 8 January 2024
• 1994 USAMO Problems/Problem 5
  ...ath>. Prove that <cmath> \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) </cmath> for all integers <math>\, m \geq \sigma(S)</math> ...ht| = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\left|\bigcap_{k \in U}A_k\right|</math>.
  3 KB (628 words) - 10:42, 4 August 2023
• Mock AIME 4 Pre 2005/Problems
  ..., q, r,</math> and <math>s</math> are positive integers such that there is no prime ...lateral in which <math>\overline{AB}\ ||\ \overline{CD}.</math> Let <math>U</math> denote the intersection of the
  7 KB (1,094 words) - 15:39, 24 March 2019
• Mock AIME 3 2010 Problems
  such that no child receives more than one piece of candy, and that no two children that sit next to each other may both ...e integers such that g, j, k and l are not divisible by any prime and e is no greater than
  14 KB (2,904 words) - 18:24, 16 May 2017
• 2018 AMC 12A Problems
  ...o that the percentage of red balls in the urn will be <math>72 \%</math>? (No red balls are to be removed.) ...at courses the student takes during the other <math>3</math> periods is of no concern here.)
  15 KB (2,380 words) - 18:52, 7 April 2022
• 2018 AIME I Problems
  ...0</math>. If <math>T</math> is chosen at random among all subsets of <math>U</math>, the probability that <math>s(T)</math> is divisible by <math>3</mat ...g stops and stays there. Find the number of distinct sequences of jumps of no more than <math>12</math> jumps that end at <math>E</math>.
  8 KB (1,284 words) - 14:35, 9 August 2021
• 2018 AIME I Problems/Problem 12
  ...0</math>. If <math>T</math> is chosen at random among all subsets of <math>U</math>, the probability that <math>s(T)</math> is divisible by <math>3</mat <cmath>U' = \{1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0\}</cmath>
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2016 UMO Problems
  ...and then removes the stick from the figure. The game ends when there are no sticks adjacent to the pebble. The player who moves last wins. A sample gam Find all positive integer pairs <math>(u,m)</math> such that <math>u + m^2</math> is divisible by <math>um-1</math>.
  3 KB (527 words) - 05:43, 20 January 2019
• 2023 AIME I Problems/Problem 10
  .../math> and <math>1000</math>. For that unique <math>a</math>, find <math>a+U</math>. First, we bound <math>U</math>.
  10 KB (1,578 words) - 09:48, 24 April 2024
• AMC 12C 2020 Problems
  ...umbered) subsets of <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</math> contain no <math>2</math> consecutive prime numbers? ...math> children if each child must receive atleast <math>1</math> candy but no child receives both types?
  8 KB (1,223 words) - 15:02, 27 November 2022
• 2020 FMC 12A Problems
  <math>u = a + bi</math> is a complex number such that <cmath>(2+i)^2 u^2 = 50i. </cmath> What is the positive value of <math>a+b?</math> ...level of authority can fire another person at a level of authority that is no greater than <math>n-2</math>. Define a firing chain as a sequence of at le
  15 KB (2,388 words) - 13:24, 9 June 2020
• 2021 AIME I Problems/Problem 14
  <li>For <math>p\equiv0\pmod{43}</math>, no matter what <math>n</math> is, it is true that <cmath>p+p^2+\cdots+p^n \equ ...}p_i^{e_i}\right)=\prod_{i=1}^{u}\sigma\left(p_i^{e_i}\right)=\prod_{i=1}^{u}\left(\sum_{j=0}^{e_i}p_i^j\right),</cmath> as this product of geometric se
  13 KB (2,185 words) - 02:28, 13 November 2023
• 1975 IMO Problems/Problem 2
  ...w apply the induction hypothesis to the sequence <math>\left(\frac{a_{k_n}}u\right)_{n\ge 1}</math>.
  1 KB (239 words) - 16:10, 29 January 2021
• 2021 AIME II Problems/Problem 9
  As <math>r</math> and <math>s</math> are relatively prime (have no prime divisors in common), this property is intuitive. ...symbol{u\geq2,}</math> then <math>\boldsymbol{\gcd\left(u^a-1,u^b-1\right)=u^{\gcd(a,b)}-1.}</math></b>
  7 KB (1,212 words) - 15:54, 15 April 2024
• 2021 USAJMO Problems/Problem 1
  ...n be proved rather simply. Note that <math>\frac{p^2-1}{2}</math> can have no prime factors greater than or equal to <math>p</math>, as it can be factore ...If <math>n=2k+1</math>, just take <math>(u,v)= (k+1,k)</math> which <math>u^2-v^2=n</math>. Thus the only answer is <math>\boxed{f\equiv 1}</math> and
  8 KB (1,492 words) - 00:23, 24 February 2024
• 2021 WSMO Speed Round Problems/Problem 5
  ...th> (A string of letters is defined as a group of consecutive letters with no spaces between them.) ...dition, the "delicious greenbeans" has 1 d, 4 e's, 1 l, 2 i's, 1 c, 1 o, 1 u, 1 g, 1 r, 2 s's, 2 n's, 1 b, and 1 a. Thus, there are <math>\frac{19!}{1!4
  1 KB (148 words) - 10:20, 23 December 2021
• Newton's method
  The function <math>f(x) = e^x</math> is always positive, so it has no roots. Since <math>f(x) = e^x</math> satisfies the differential equation <m ...for <math>y\in\mathbb{Z}_p</math> and <math>p'(x)=p^ku</math> where <math>u\in\mathbb{Z}_p^{\times}</math>. By definition we see that
  13 KB (2,298 words) - 23:34, 28 May 2023
• 2023 AIME I Problems
  ...t, <math>6</math> points where exactly <math>6</math> lines intersect, and no points where more than <math>6</math> lines intersect. Find the number of p .../math> and <math>1000</math>. For that unique <math>a</math>, find <math>a+U</math>.
  7 KB (1,154 words) - 12:54, 20 February 2024
• 2023 AIME I Problems/Problem 14
  path u=(3,8)--(4,8)--(4,9)--(4,10)--(4,11)--(5,11)--(5,11.5); draw(u,p);
  11 KB (1,834 words) - 22:01, 4 January 2024
• 2023 AIME I Problems/Problem 13
  pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v));
  13 KB (2,130 words) - 01:52, 31 January 2024
• Pell's equation (simple solutions)
  b) to prove that the equation has no other integer solutions. There is no integer solution if <math> y_j = 1. y_j = 0</math> is impossible. So <math>
  15 KB (2,351 words) - 08:28, 20 April 2023
• 2023 AMC 10B Problems/Problem 16
  ...r <math>258</math> is an upno and <math>8620</math> is a downno. Let <math>U</math> equal the total ...th>D</math> equal the total number of <math>downnos</math>. What is <math>|U-D|</math>?
  8 KB (1,195 words) - 12:03, 26 February 2024
• 2023 AMC 12B Problems/Problem 25
  pair K, L, M, N, O, P, Q, R, S, T, U, V; pair K, L, M, N, O, P, Q, R, S, T, U, V;
  19 KB (2,967 words) - 16:56, 24 February 2024
• 2024 AMC 8 Problems/Problem 13
  ...ber of ways to insert 2 <math>U</math>'s and 2 <math>D</math>'s into <math>U\, \_ \, \_ \, \_ \, \_ \, D</math>. There are <math>{4\choose 2}=6</math> w ...<math>D</math>'s (since we removed the one case), and at least three <math>U</math>'s before three <math>D</math>'s, as we end with the third <math>D</m
  4 KB (683 words) - 09:40, 19 February 2024
• 2024 AIME I Problems/Problem 6
  ...vide the path into eight “<math>R</math>” movements and eight “<math>U</math>” movements. Five sections of alternative <math>RURUR</math> or <ma For <math>U</math>, we have seven ordered pairs of positive integers <math>(a,b)</math>
  6 KB (932 words) - 10:44, 10 February 2024
• Proofs of trig identities
  Pythagorean identities are easy and there's no algebra involved. In fact, the name Pythagorean is a giveaway of what we sh Then, <math>u=\frac{-4y\pm\sqrt{16y^2-1}}{8}</math>
  16 KB (2,796 words) - 13:12, 21 January 2024
• DVI exam
  Denote <math>u = \frac {x}{4}, v = \frac {t}{4}.</math> <cmath>\left\{\begin{array}{l} u^{\frac{3}{2}} + v^{\frac{3}{2}} = 2 ,\\u + v = 2.\end{array}\right.</cmath>
  31 KB (5,191 words) - 08:10, 30 April 2024